Abstract
We study the local geometry of 4-manifolds equipped with a para-Kähler-Einstein (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated twistor distribution, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with non-zero scalar curvature this twistor distribution has exactly two integral leaves and is ‘maximally non-integrable’ on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-zero scalar curvature and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartan’s method of equivalence to produce a large number of explicit examples of pKE metrics with non-zero scalar curvature whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type D, we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding Cartan connections satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.
Highlights
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The para-Kähler condition is used to reduce the structure equations of an almost para-Hermitian structure. After deriving their structure equations and curvature decomposition, we show that para-Kähler structures can be described in terms of a potential function, using which we give two examples of para-Kähler-Einstein metrics
This section is the heart of the article, in which we describe pKE structures as Cartan geometries, give an in-depth study when the Petrov type is real and special and provide explicit examples
Summary
The goal of this section is to fix notation, give necessary definitions and recall some facts that will be needed in subsequent sections. 2.1 we recall some basic facts about pseudo-Riemannian metrics in dimension four. The notion of almost para-Hermitian structure and the decomposition of its curvature into irreducible components with respect to the action of its structure group is defined in Sect. 2.3 we define para-Kähler structures in terms of additional integrability and compatibility conditions imposed on an almost paraHermitian structure. We derive their structure equations, curvature decomposition and give a local coordinate expression in terms of a potential function. We end the section by giving examples of pKE structures in terms of potential functions which, as will be shown in Sect. 3.2, correspond to the only homogeneous models with non-zero scalar curvature
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